Twelve points on the projective line, branched covers, and rational elliptic fibrations

Abstract.

The following divisors in the space ${\rm Sym}^{12}{\mathbb P}^1$ of twelve points on ${\mathbb P}^1$ are actually the same: $({\mathcal A})$ the possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); $({\mathcal B})$ degree 12 binary forms that can be expressed as a cube plus a square; $({\mathcal C})$ the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; $({\mathcal D})$ the branch locus of a degree 4 map from a hyperelliptic genus 3 curve to ${\mathbb P}^1$; $({\mathcal E})$ the branch locus of a degree 3 map from a genus 4 curve to ${\mathbb P}^1$ induced by a theta-characteristic; and several more. The corresponding moduli spaces are smooth, but they are not all isomorphic; some are finite étale covers of others. We describe the web of interconnections among these spaces, and give monodromy, rationality, and Prym-related consequences. Enumerative consequences include: (i) the degree of this locus is 3762 (e.g. there are 3762 rational elliptic fibrations with nodes above 11 given general points of the base); (ii) if $C \rightarrow{\mathbb P}^1$ is a cover as in $({\mathcal D})$, then there are 135 different such covers branched at the same points; (iii) the general set of 12 tangent lines that arise in $({\mathcal C})$ turn up in 120 essentially different ways. Some parts of this story are well known, and some other parts were known classically (to Zeuthen, Zariski, Coble, Mumford, and others). The unified picture is surprisingly intricate and connects many beautiful constructions, including Recillas' trigonal construction and Shioda's $E_8$-Mordell-Weil lattice.